Finite Group Databasebeta

← #763117

Subgroups — #763117

Order 512.

2420 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#126C211yesmaximal, characteristic
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic
5#16560C211yesmaximal, characteristic, complemented
6#16511C211yesmaximal, characteristic
7#15206C212yesmaximal, complemented
8#15204C212yesmaximal
9#15105C212yesmaximal, complemented
10#15103C212yesmaximal
11#14872C212yesmaximal, complemented
12#14866C212yesmaximal
13#14242C212yesmaximal, complemented
14#14230C212yesmaximal
15#14174C212yesmaximal, complemented
16#14172C212yesmaximal
17#14166C212yesmaximal
18#13952C212yesmaximal, complemented
19#13755C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#395C2212yes—
26C2×#393C2212yes—
27C4×C4⋊Q8—28no—
28C4×#126—28no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2211yescharacteristic
33C2×#388C2211yescharacteristic
34C2×#388C2211yescharacteristic
35C2×#387C2211yescharacteristic
36C2×#387C2211yescharacteristic
37C2×#387C2211yescharacteristic
38C2×#387C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#383C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×#376C2212yes—
52C2×C43C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2212yes—
77#1437C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85C4⋊#95C2212yes—
86C4⋊#95C2214yes—
87C4⋊#95C2212yes—
88#1434C2214yes—
89#1434C2212yes—
90#1434C2211yescharacteristic
91#1434C2211yescharacteristic
92C4×#95C2212yes—
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#126Q818yescomplemented
100C2×C4×Q8—28no—
101C2×C4×Q8—48no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#400—28no—
138#399—28no—
139#395—28no—
140#393—28no—
141#391—28no—
142#389—28no—
143#389—28no—
144#388—24no—
145#388—24no—
146#388—24no—
147#387—24no—
148#387—24no—
149#387—24no—
150#387—48no—
151#385—48no—
152#384—24no—
153#383—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—28no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163#376—28no—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C43D418yes—
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—28no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C4⋊Q8—232no—
228#126—232no—
229C4×Q8—432no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×Q814yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4#4118yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×Q814yes—
287C2×C22⋊C4C2×Q814yes—
288C2×C22⋊C4C2×Q814yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×Q814yes—
315C2×C42#4114yes—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95#4118yes—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24C22×Q812yes—
359C24C2511yescharacteristic
360C2×Q8—216no—
361C2×Q8—416no—
362C2×Q8—416no—
363C2×Q8—48no—
364C2×Q8—48no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—88no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C42−1+414yes—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—416no—
472C4⋊C4—44no—
473C4⋊C4—88no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—416no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—88no—
482C42—28no—
483C42—28no—
484C42—28no—
485C42—28no—
486C42—44no—
487C42—44no—
488C42—416no—
489C42—44no—
490C42—44no—
491C42—416no—
492C42—44no—
493C42C22⋊Q8116yes—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—44no—
511C23C23×D411yescharacteristic
512C23C23×Q811yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4112yes—
515C23C22×#4112yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+412yes—
518C23#58512yes—
519C23#58512yes—
520Q8—432no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—88no—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q814yes—
607C2×C4C2×C22⋊Q814yes—
608C2×C4#52514yes—
609C2×C4#53214yes—
610C2×C4#53514yes—
611C2×C4#53614yes—
612C22—48no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—84no—
620C22—81no—
621C22C22×C22⋊Q811yescharacteristic
622C22C2×#52511yescharacteristic
623C22C2×#53211yescharacteristic
624C22C2×#53511yescharacteristic
625C22C2×#53611yescharacteristic
626C22#344911yescharacteristic
627C22#344911yescharacteristic
628C22#347512yes—
629C22Q8×#4112yes—
630C22Q8×#4112yes—
631C22#348012yes—
632C22#348114yes—
633C22#349514yes—
634C22#350312yes—
635C22#350714yes—
636C22#351112yes—
637C22#356014yes—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—48no—
648C4—42no—
649C4—48no—
650C4—81no—
651C4—81no—
652C4—84no—
653C4—81no—
654C4—84no—
655C4—81no—
656C2—82no—
657C2#3494711yescharacteristic
658C2#3495211yescharacteristic
659C2#3496511yescharacteristic
660C2#3511111yescharacteristic
661C2#3527911yescharacteristic
662C2#3532611yescharacteristic
663C2#3754711yescharacteristic
664C2#5990014yes—
665C2#5993214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C4×C4⋊Q8C2semidirect
3. #17067C2semidirect
5. #16560C2semidirect
7. #15206C2semidirect
9. #15105C2semidirect
11. #14872C2semidirect
13. #14242C2semidirect
15. #14174C2semidirect
18. #13952C2semidirect
19. #13755C2semidirect
99. C2×#126Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×C4⋊Q82
C2×C4×#1262
#170672
#170662
#165602
#165112
#152062
#152042
#151052
#151032
#148722
#148662
#142422
#142302
#141742
#141722
#141662
#139522
#137552